Give an example of:
(a) a physical quantity which has a unit but no dimensions
(b) a physical quantity which has neither unit nor dimensions
(c) a constant which has a unit
(d) a constant which has no unit
(a) Plane angle $ \theta = \frac{L}{r} \text{radian}$
its unit is radian but has no dimensions
(b) Strain $ = \frac{\Delta L}{L} = \frac{\text{Change in length}}{\text{length}}$
It has neither unit nor dimensions
(c) Gravitational constant (G) = $6.67 \times 10^{-11} \text{N} m^{2} / \text{kg}^{2}$
(d) Reynold’s number is a constant which has no unit.
Calculate the length of the arc of a circle of radius $31.0 \text{ cm}$ which subtends an angle of $ \frac{ \pi}{6}$ at the centre.
We know that angle $ \theta = \frac{l}{r} \text{radian}$
Given, $ \theta = \frac{ \pi}{6} = \frac{l}{31 \text{ cm}}$
Hence, length $l = 31 \times \frac{ \pi}{6} \text{ cm} = \frac{31 \times 3.14}{6} \text{ cm} = 16.22 \text{ cm}$
Rounding off to three significant figures it would be $16.2 \text{ cm}$.
Calculate the solid angle subtended by the periphery of an area of $1\text{cm}^{2}$ at a point situated symmetrically at a distance of $5 \text{ cm}$ from the area.
We know that solid angle $ \Omega = \frac{\text{Area}}{(\text{Distance})^{2}}$
$= \frac{1 \text{cm}^{2}}{(5 \text{cm})^{2}}= \frac{1}{25} = 4 \times 10^{-2} \text{ steradian}$
(∵ Area = $1 \text{cm}^{2}$, distance = $5 \text{ cm}$)
Note We should not confuse, solid angle with plane angle $ \theta = \frac{l}{r} \text{ radian}$.
$y = A \sin(\omega t - kx)$, where $x$ is distance and $t$ is time. Write the dimensional formula of (i) $\omega$ and (ii) $k$.
Now, by the principle of homogeneity, i.e., dimensions of LHS and RHS should be equal, hence
$[LHS] = [RHS]$
$$ \Rightarrow $$$[L] = [A] = L$
As $\omega t - kx$ should be dimensionless, $[\omega t] = [kx] = 1$
$$ \Rightarrow $$$[\omega] T = [k] L = 1$
$$ \Rightarrow $$$[\omega] = T^{-1}$ and $[k] = L^{-1}$
Time for 20 oscillations of a pendulum is measured as $t_1 = 39.6$ s; $t_2 = 39.9$ s and $t_3 = 39.5$ s. What is the precision in the measurements? What is the accuracy of the measurement?
Given, $t_1 = 39.6$ s, $t_2 = 39.9$ s and $t_3 = 39.5$ s
Least count of measuring instrument = 0.1 s
(As measurements have only one decimal place)
Precision in the measurement = Least count of the measuring instrument = 0.1 s
Mean value of time for 20 oscillations is given by
$t = \frac{t_1 + t_2 + t_3}{3} = \frac{39.6 + 39.9 + 39.5}{3} = 39.7\text{ s}$
Absolute errors in the measurements
$\Delta t_1 = t - t_1 = 39.7 - 39.6 = 0.1\text{ s}$
$\Delta t_2 = t - t_2 = 39.7 - 39.9 = -0.2\text{ s}$
$\Delta t_3 = t - t_3 = 39.7 - 39.5 = 0.2\text{ s}$
Mean absolute error =
$\frac{|\Delta t_1| + |\Delta t_2| + |\Delta t_3|}{3} = \frac{0.1 + 0.2 + 0.2}{3} = \frac{0.5}{3} \approx 0.2\text{ s} \quad\text{(rounding off up to one decimal place)}$
$\therefore$ Accuracy of measurement = $\pm 0.2\text{ s}$